To provide a little bit of context to the above question, it's not so hard to show that the statement is implied by the axiom of choice (first replace $A$ with an ordinal, then build a bijection by transfinite induction).
Also, if you replace the statement by the stronger statement "for every two infinite sets $A$ and $B$ there is a bijection $A\coprod B\cong A$ or a bijection $A\coprod B\cong B$" then the statement indeed implies the axiom of choice: given an infinite set $A$, use Hartog's lemma to find a well ordered set $B$ such that $B$ doesn't inject into $A$, then by the above statement we have that $A$ injects into $B$ and $A$ acquires a well-ordering.
But what is less clear (to me) is that the statement implies the axiom of choice. Does it?
As an aside, I see the site is awash with people asking the question "is ... equivalent to the axiom of choice" and have tried to check that this question isn't a duplicate. Apologies if it is.
